Influence functions for iteratively defined statistics

Abstract

SUMMARY Hampel's influence function and its finite-sample counterparts are the basis for a number of diagnostic statistics. These diagnostics can be expensive to compute in the natural way when the estimation calculations are iterative, as they frequently are when maximum likelihood or robust methods are used. We show how the influence function can be calculated in these situations by implicit differentiation of the fixed-point equation satisfied by the limit of the iterative process. We consider in particular the cases of Newton's method and iteratively reweighted least squares where interesting analytic results are available. As an application we consider the generalization of Pregibon's (1981) logistic regression diagnostics to cover generalized linear models with noncanonical link functions such as probit regression.